How to Calculate Distance

Distance is a fundamental concept in mathematics and physics, representing the extent of space between two points, lines, or planes. It's a crucial metric used in various fields, including navigation, physics, engineering, and everyday life. Calculating distance accurately is essential for solving problems related to travel, motion, and spatial relationships. In this article, we will explore different methods to calculate distance between points, lines, and planes.

What is Distance?

Distance is a numerical measurement of how far apart objects or points are from each other in space.

It is a fundamental concept in mathematics, physics, and everyday life, used to quantify the extent of separation between two entities. Distance can be measured along a straight line or path between two points, or it can be more complex, such as the distance along a curved surface or in a multidimensional space.

How to Calculate Distance?

Below, we will calculate distance in different ways such as:

• Distance, Speed, and Time Formula
• Distance Between Two Points Using Coordinates
• Distance Between Two Lines
• Distance Between Two Planes

Let's discuss each case in detail as follows.

Distance, Speed, and Time Formula

One of the fundamental formulas used to calculate distance involves the relationship between distance, speed, and time.

The distance formula is given by:

Distance = Speed × Time

This formula expresses that the distance traveled is equal to the product of the speed at which an object is moving and the time it has been moving at that speed.

Distance Between Two Points Using Coordinates

The formula to calculate the distance between two points using coordinates (x₁, y₁) and (x₂, y₂) is given by:

Distance = √ [(x₂ - x₁)² + (y₂ - y₁)²]

Here, (x₁, y₁) and (x₂, y₂) are the coordinates of the two points. This distance formula is also known as Pythagorean theorem.

Distance Between Two Lines

The distance between two lines can be determined between two parallel or two non-parallel lines.

Distance Formula for Two Parallel Lines

The formula to calculate the distance between two parallel lines when equations are of the form y = mx + c₁ and y = mx + c₂ respectively is given by:

Distance = \bold{\frac{|c_1 - c_2|}{\sqrt{1 + m^2}}}

The formula to calculate the distance between two parallel lines when equations are of the form ax + by + c₁ = 0 and ax + by + c₂ = 0 respectively is given by:

Distance = \bold{\frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}}

Distance Formula for Two Non-Parallel Lines

The formula to calculate the distance between two non- parallel lines whose equations are \overrightarrow{\rm r_1} = \overrightarrow{\rm a_1}+t\overrightarrow{\rm b_1} and \overrightarrow{\rm r_2} = \overrightarrow{\rm a_2}+t\overrightarrow{\rm b_2} respectively is given by:

\bold{D = \frac{(\overrightarrow{\rm a_2}-\overrightarrow{\rm a_1}).(\overrightarrow{\rm b_2}-\overrightarrow{\rm b_1})}{(\overrightarrow{\rm b_1}\times \overrightarrow{\rm b_2})}}

Distance Between Two Planes

In three-dimensional space, calculating the distance between two planes involves finding the shortest distance between any two points on the planes. This can be done using vector algebra and the dot product.

The formula to find the distance between two planes is:

\text{Distance} = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}

Here, ax + by + cz + d = 0 represents the equation of the plane, and (x₁, y₁, z₁) is a point on the plane.

Conclusion

In conclusion, distances can be calculated using various formulas discussed in this article. Distances between two points can be calculated if given as (x₁, y₁, z₁) and (x₂, y₂, z₂), or distances between two lines, whether parallel or not, can also be determined. Additionally, distances between planes in 3D can also be calculated.

Read More,

• 3D Distance Formula
• Euclidean Distance
• Distance and Displacement
• Coordinate Geometry

Solved Examples on How to Calculate Distance

Example 1: Calculate the distance given the speed is 20 m/s and time is 10 s.

Solution:

To calculate distance, we use below formula.

Distance = Speed × Time

Distance = 20 m/s × 10s

Distance = 200 m

Example 2: Calculate the distance between two points with coordinates (5, 3) and (8, 2).

Solution:

To calculate distance, we use below formula.

Distance = √ [(x₂ - x₁)² + (y₂ - y₁)²]

Here,

(x₁, y₁) = (5, 3) and (x₂, y₂) = (8, 2)

Distance = √ [(8 - 5)² + (2 - 3)²]

Distance = √ [(3)² + (-1)²]

Distance = √ [9 + 1]

Distance = √10 units

Example 3: Find the distance between two parallel lines with equations y1 = 2x + 10 and y2 = 2x + 15.

Solution:

To find distance we use formula

Distance = \bold{\frac{|c_1 - c_2|}{\sqrt{1 + m^2}}}

Distance = |10 - 15| / (1 + 2²)

Distance = 5 / (1 + 4)

Distance = 5 / 5

Distance = 1

Example 4: Find the distance between two planes whose equations are 2x + 3y + 4z + 5 = 0 and 2x + 3y + 4z + 10 = 0.

Solution:

To find distance between two planes

Distance = \bold{\frac{|d_1 - d_2|}{\sqrt{a^2 + b^2+c^2}}}

Distance = |5 - 10| / √(2² + 3² + 4²)

Distance = 5 / √(4 + 9 + 16)

Distance = 5 / √29

Practice Problems on How to Calculate Distance

Q1. Calculate the distance given the speed is 45 m/s and time is 5 s.

Q2. Calculate the distance between two points with coordinates (10, 15) and (20, 25).

Q3. Calculate the distance between two parallel lines with equation 3x + 9y + 10 = 0 and 4x + 2y + 3 = 0.

Q4. Calculate the distance between two non-parallel lines with equations \overrightarrow{\rm r_1} = 2\hat{i} + 3\hat{j}+t(4\hat{i} + 7\hat{j}) and \overrightarrow{\rm r_1} = 4\hat{i} - 6\hat{j}+t(9\hat{i} - 2\hat{j})

Q5. Find the distance between two planes whose equations are 12x + 7y + 9z + 3 = 0 and 5x + 13y + 6z + 15 = 0.